Math 253 - Homework 2 Due in class on Wednesday, February 19
Write your answers clearly and carefully, being sure to emphasize your answer and the key steps of your work. You may work with others in this class, but the solutions handed in must be your own. If you work with someone or get help from another source, give a brief citation on each problem for which that is the case.
Part I
While you are expected to complete all of these problems, do not hand in the problems in Part I. You are encouraged to write complete solutions and to discuss them with me or your peers. As extra motivation, some of these problems will appear on the weekly quizzes. 1. Practice Problems: (a) Section 1.3: 1, 2 2. Exercises: (a) Section 1.3: 1-13 odd, 19, 21, 25, 27, 29, 33
Part II
Hand in each problem separately, individually stapled if necessary. Please keep all problems together with a paper clip.
n 1. The center of mass of v1,..., vk in R , with a mass of mi at vi, for i = 1, . . . , k and total mass m = m1 + ··· + mk, is given by m v + ··· + m v m m v¯ = 1 1 k k = 1 v + ··· + k v . m m 1 m k n The centroid, which can be thought of as the geometric center, of v1,..., vk in R is the center of mass with mi = 1 for each i = 1, . . . , k, i.e. v + ··· + v 1 1 c¯ = 1 k = v + ··· + v . k k 1 k k 4 −1 1 −5 1 Pk (a) Let v1 = 1 , v2 = 0 , v3 = 3 , v4 = 0 , v5 = −4 . Let uk = i=1 vi. Compute 1 5 −3 2 −5 each ui for i = 1,..., 5.
(b) Compute the centroid of the set {v1,..., v5} and again of the set {u1,..., u5}. Explain why the the centroid of the vi’s might have been expected.
(c) Interpret the centroid of the ui’s as a linear combination of the vi’s. Is this realizable as the center of mass of the vi’s for some masses mi?
(d) Set up and solve a system of equations to find a weight assignment for v1, v2, v3 such that the 1 center of mass of these three vectors is at v¯ = 1 . How many solutions to the system are there? 1 1 Does this accurately describe the number of possible masses which will produce v¯ = 1? Explain 1 your reasoning. 1 −2 a 2. (a) Let u = and v = . Show that for any a and b, the vector is in Span{u, v}. −5 2 b 1 −2 777 (b) Let u = and v = . For which values of a is it true that the vector is in −5 a 9311 Span{u, v}? Interpret your answer geometrically. 1 −2 3 (c) Let u = and v = . For which values of a and b is it true that the vector is in −5 a b Span{u, v}? 1 cos θ (d) Let u = and v = . Give an explicit description of Span{u, v} in terms of θ and interpret 0 sin θ your answer geometrically. 3. (a) A system of linear equations is underdetermined if it has fewer equations than variables. Discuss the possible sizes of the solution set for an underdetermined system; for each possibility, give an augmented matrix in reduced echelon form which illustrates your claim. (b) A system of linear equations is overdetermined if it has more equations than variables. Discuss the possible sizes of the solution set for an overdetermined system; for each possibility, give an augmented matrix in reduced echelon form which illustrates your claim. (c) A system of linear equations is homogeneous if each equation has a constant term of zero (i.e. each entry in the augmentation column of the corresponding matrix is zero). Discuss the possible sizes of the solution set for a homogeneous system; for each possibility, give an augmented matrix in reduced echelon form which illustrates your claim. 4. A common way to describe a line in 3-dimensions is with parametric equations for x, y and z. Consider the system of parametric equations which define a line `(t): x(t) = 3t − 4 `(t) = y(t) = −t − 1 z(t) = 2t + 1 where t is any real number. (a) Find a system of two equations in three unknowns which represents two distinct planes whose intersection is `. (b) Find a system of three equations in three unknowns which represents three distinct planes whose intersection is `. These planes should be all be distinct from your answers from Part (a). (c) Select one equation from your answer in Part (a) and one equation from your answer in Part (b). Make an educated guess as to what the intersection of their corresponding planes should be. Explain. (d) Verify your guess in part (b) by finding the intersection of the pair of planes you chose. 5. Consider the vectors 1 1 0 0 4 1 −1 0 2 0 v1 = , v2 = , v3 = , v4 = , and b = . 0 1 1 0 0 0 0 1 1 −2
(a) Show that b is a member of Span{v1, v2, v3, v4}. Are the weights for your linear combination unique? In other words, is there only one choice of scalars c1, c2, c3, and c4 for which c1v1 + c2v2 + c3v3 + c4v4 = b?
(b) Show that b is a member of Span{v1, v2, v3}. Are the weights for your linear combination unique?
Page 2 (c) To understand the distinction between the two preceding items, we introduce an important concept which will follow us throughout the course:
n Definition 1. Let u1, u2,..., up be vectors in R and let 0 denote the zero vector. We say that the vectors u1, u2,..., up are linearly independent if the equation
c1u1 + c2u2 + ··· + cpup = 0
can only be satisfied if c1, c2, . . . , cp are all zero.
1 Show that the vectors v1, v2, v3, v4 are not linearly independent .
(d) Show that the vectors v1, v2, v3 are linearly independent. n (e) Returning to the general picture, let u1, u2,..., up be vectors in R and let b ∈ Span{u1, u2,..., up}. Prove (i.e. give a convincing argument for) the following proposition.
Proposition 1. If u1, u2,..., up are linearly independent, then there is a unique (only one) choice of weights c1, c2, . . . , cp, for which
b = c1u1 + c2u2 + ··· + cpup.
Hint: You may want to suppose that b can be expressed using two (possibly) different sets of weights c1, c2, . . . , cp and a1, a2, . . . , ap. Then use linear independence to show that the weights all must be the same. (f) Finally, explain why your conclusion for Part (a) was different from your conclusion for Part (b). Your explanation should be no more than two sentences and it should reference the proposition above.
m 6. Given vectors a1, a2,..., an and b in R , consider the matrix equation x1 x2 Ax = [a a ··· a ] = b. (1) 1 2 n . . xn
We note, in view of the definition on Page 41 of the text, the matrix product above is defined to be x1 x2 Ax = [a a ··· a ] = x a + x a + ··· + x a . 1 2 n . 1 1 2 2 n n . xn x1 . (a) Show that Equation (1) has a solution x = . if and only if b ∈ Span{a1, a2,..., an}. xn (b) In view of the previous question (specifically the proposition therein), formulate a precise (and true) statement concerning the uniqueness of solutions to Equation (1) and linear independence/dependence of the vectors a1, a2,..., an.
1Unsurprisingly, such a collection is said to be linearly dependent.
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